External Direct Product Commutativity/Sufficient Condition

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Theorem

Let $\struct {S \times T, \circ}$ be the external direct product of the two algebraic structures $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.

Let $\circ_1$ and $\circ_2$ be commutative operations.


Then $\circ$ is also a commutative operation.


General Result

Let $\ds \struct {S, \circ} = \prod_{k \mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$.


If $\circ_1, \ldots, \circ_n$ are all commutative, then so is $\circ$.


Proof

Let $\circ_1$ and $\circ_2$ be commutative operations.

\(\ds \tuple {s_1, t_1} \circ \tuple {s_2, t_2}\) \(=\) \(\ds \tuple {s_1 \circ_1 s_2, t_1 \circ_2 t_2}\) Definition of Operation Induced by Direct Product
\(\ds \) \(=\) \(\ds \tuple {s_2 \circ_1 s_1, t_2 \circ_2 t_1}\) Definition of Commutative Operation
\(\ds \) \(=\) \(\ds \tuple {s_2, t_2} \circ \tuple {s_1, t_1}\) Definition of Operation Induced by Direct Product

Thus $\circ$ is commutative.

$\blacksquare$


Also see

Sources